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Theorem uniqsALTV 34551
Description: The union of a quotient set: a weaker version of uniqs 8014. (Contributed by Peter Mazsa, 20-Jun-2019.)
Assertion
Ref Expression
uniqsALTV ((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

Proof of Theorem uniqsALTV
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecex2 34550 . . . . 5 ((𝑅𝐴) ∈ 𝑉 → (𝑥𝐴 → [𝑥]𝑅 ∈ V))
21ralrimiv 3112 . . . 4 ((𝑅𝐴) ∈ 𝑉 → ∀𝑥𝐴 [𝑥]𝑅 ∈ V)
3 dfiun2g 4710 . . . 4 (∀𝑥𝐴 [𝑥]𝑅 ∈ V → 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
42, 3syl 17 . . 3 ((𝑅𝐴) ∈ 𝑉 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
54eqcomd 2771 . 2 ((𝑅𝐴) ∈ 𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} = 𝑥𝐴 [𝑥]𝑅)
6 df-qs 7957 . . 3 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
76unieqi 4605 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
8 df-ec 7953 . . . . 5 [𝑥]𝑅 = (𝑅 “ {𝑥})
98a1i 11 . . . 4 (𝑥𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
109iuneq2i 4697 . . 3 𝑥𝐴 [𝑥]𝑅 = 𝑥𝐴 (𝑅 “ {𝑥})
11 imaiun 6699 . . 3 (𝑅 𝑥𝐴 {𝑥}) = 𝑥𝐴 (𝑅 “ {𝑥})
12 iunid 4733 . . . 4 𝑥𝐴 {𝑥} = 𝐴
1312imaeq2i 5648 . . 3 (𝑅 𝑥𝐴 {𝑥}) = (𝑅𝐴)
1410, 11, 133eqtr2ri 2794 . 2 (𝑅𝐴) = 𝑥𝐴 [𝑥]𝑅
155, 7, 143eqtr4g 2824 1 ((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350  {csn 4336   cuni 4596   ciun 4678  cres 5281  cima 5282  [cec 7949   / cqs 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-xp 5285  df-rel 5286  df-cnv 5287  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ec 7953  df-qs 7957
This theorem is referenced by:  rnresequniqs  34552
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